Investigation of the parallel efficiency of a PC cluster for the simulation of a CFD problem

Abstract

Previously, large-scale fluid dynamics problem required supercomputers, such as the Cray, and took a long time to obtain a solution. Clustering technology has changed the world of the supercomputer and fluid dynamics. Affordable cluster computers have replaced the huge and expansive supercomputers in computational fluid dynamics (CFD) field in recent years. Even supercomputers are designed in the form of clusters based on high-performance servers. This paper describes the configuration of the affordable PC hardware cluster as well as the parallel computing performance using commercial CFD code in the developed cluster. A multi-core cluster using the Linux operating system was developed with affordable PC hardware and low-cost high-speed gigabit network switches instead of Myrinet or Infiniband. The PC cluster consisted of 52 cores and easily expandable up to 96 cores in the current configuration. For operating software, the Rock cluster package was installed in the master node to minimize the need for maintenance. This cluster was designed to solve large fluid dynamics and heat transfer problems in parallel. Using a commercial CFD package, the performance of the cluster was evaluated by changing the number of CPU cores involved in the computation. A forced convection problem around a linear cascade was solved using the CFX program, and the heat transfer coefficient along the surface of the turbine cascade was simulated. The mesh of the model CFD problem has 1.5 million nodes, and the steady computation was performed for 2,000 time-integrations. The computation results were compared with previously published heat transfer experimental results to check the reliability of the computation. A comparison of the simulation and experimental results showed good agreement. The performance of the designed PC cluster increased with increasing number of cores up to 16 cores The computation (elapsed) 16-core was approximately three times faster than that with a 4-core.

Appendix

1.1 The coefficients of the shear stress transport model

The set of empirical constants Φ = (σ _k , σ _ω , β, γ) used in the baseline model was calculated from two sets of constants Φ₁ and Φ₂ as follows:

$$\Upphi = \, F_{1} \Upphi_{1} + \left( {1 - F_{1} } \right)\Upphi_{2}$$

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where the set of constants, Φ₁, was derived from the original k − ω model such that

$$\sigma_{\text{k1}} = 0. 5,\;\sigma_{\omega 1} = 0. 5,\;\beta_{ 1} = 0.0 7 5,\;\beta^{*} = 0.0 9,\;\gamma_{1} = \beta_{1} /\beta^{*} - \frac{{\sigma_{\omega 1} \kappa^{2} }}{{\sqrt {\beta^{*} } }}$$

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and the set of constants, k − ω, was derived from the standard k − ω model such that

$$\sigma_{\text{k2}} { = 1}.0,\;\sigma_{\omega 2} { = 1}. 8 5 6,\;\beta_{ 2} = 0.0 8 2 8,\;\beta^{*} = 0.0 9 ,\;\gamma_{2} = \beta_{2} /\beta^{*} - \frac{{\sigma_{\omega 2} \kappa^{2} }}{{\sqrt {\beta^{*} } }}.$$

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F ₁ can be expressed as

$$F_{1} = \tanh (\arg_{1}^{4} ),\;\arg_{1} = \hbox{min} \left[ {\hbox{max} \left( {\frac{\sqrt k }{0.00\omega y};\frac{500\nu }{{y^{2} \omega }}} \right);\frac{{4\rho \sigma_{\omega 2} k}}{{CD_{k\omega } y^{2} }}} \right]$$

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where y is the distance to the next surface and CD _kω is the positive portion of the cross-diffusion term

$$CD = \hbox{max} \left( {2\rho \sigma_{\omega } \frac{1}{\omega }\frac{\partial k}{{\partial x_{j} }}\frac{\partial \omega }{{\partial x_{j} }};10^{ - 20} } \right).$$

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The eddy viscosity is defined as

$$\nu_{t} = \frac{{a_{1} k}}{{\hbox{max} (a_{1} \omega ;\Upomega F_{2} )}}$$

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where Ω is the absolute value of the vorticity. F ₂ can be expressed as

$$\begin{gathered} F_{2} = \tanh (\arg_{2}^{2} ) \hfill \\ \arg_{2} = \hbox{max} \left( {\frac{\sqrt k }{0.09\omega y};\frac{500\nu }{{y^{2} \omega }}} \right) \hfill \\ \end{gathered}.$$

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